Optimal. Leaf size=61 \[ \frac {2 \sqrt {a-b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^2}+\frac {b \tan ^{-1}(\sinh (x))}{a^2}-\frac {\tanh (x)}{a} \]
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Rubi [A] time = 0.23, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2723, 3056, 3001, 3770, 2659, 208} \[ \frac {2 \sqrt {a-b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^2}+\frac {b \tan ^{-1}(\sinh (x))}{a^2}-\frac {\tanh (x)}{a} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2659
Rule 2723
Rule 3001
Rule 3056
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tanh ^2(x)}{a+b \cosh (x)} \, dx &=-\int \frac {\left (1-\cosh ^2(x)\right ) \text {sech}^2(x)}{a+b \cosh (x)} \, dx\\ &=-\frac {\tanh (x)}{a}-\frac {\int \frac {(-b-a \cosh (x)) \text {sech}(x)}{a+b \cosh (x)} \, dx}{a}\\ &=-\frac {\tanh (x)}{a}+\frac {b \int \text {sech}(x) \, dx}{a^2}-\frac {\left (-a^2+b^2\right ) \int \frac {1}{a+b \cosh (x)} \, dx}{a^2}\\ &=\frac {b \tan ^{-1}(\sinh (x))}{a^2}-\frac {\tanh (x)}{a}-\frac {\left (2 \left (-a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^2}\\ &=\frac {b \tan ^{-1}(\sinh (x))}{a^2}+\frac {2 \sqrt {a-b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^2}-\frac {\tanh (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 61, normalized size = 1.00 \[ \frac {2 \sqrt {b^2-a^2} \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )-a \tanh (x)+2 b \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )}{a^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 326, normalized size = 5.34 \[ \left [\frac {\sqrt {a^{2} - b^{2}} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) + b}\right ) + 2 \, {\left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} + b\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + 2 \, a}{a^{2} \cosh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) \sinh \relax (x) + a^{2} \sinh \relax (x)^{2} + a^{2}}, -\frac {2 \, {\left (\sqrt {-a^{2} + b^{2}} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{a^{2} - b^{2}}\right ) - {\left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} + b\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) - a\right )}}{a^{2} \cosh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) \sinh \relax (x) + a^{2} \sinh \relax (x)^{2} + a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 67, normalized size = 1.10 \[ \frac {2 \, b \arctan \left (e^{x}\right )}{a^{2}} + \frac {2 \, {\left (a^{2} - b^{2}\right )} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} a^{2}} + \frac {2}{a {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 108, normalized size = 1.77 \[ \frac {2 \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {2 \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right ) b^{2}}{a^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}+\frac {2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right ) b}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 285, normalized size = 4.67 \[ \frac {2}{a+a\,{\mathrm {e}}^{2\,x}}+\frac {\ln \left (64\,a^3\,b-64\,a\,b^3-32\,b^3\,\sqrt {a^2-b^2}+128\,a^4\,{\mathrm {e}}^x+32\,b^4\,{\mathrm {e}}^x+64\,a^2\,b\,\sqrt {a^2-b^2}+128\,a^3\,{\mathrm {e}}^x\,\sqrt {a^2-b^2}-160\,a^2\,b^2\,{\mathrm {e}}^x-96\,a\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2-b^2}\right )\,\sqrt {a^2-b^2}}{a^2}-\frac {\ln \left (64\,a^3\,b-64\,a\,b^3+32\,b^3\,\sqrt {a^2-b^2}+128\,a^4\,{\mathrm {e}}^x+32\,b^4\,{\mathrm {e}}^x-64\,a^2\,b\,\sqrt {a^2-b^2}-128\,a^3\,{\mathrm {e}}^x\,\sqrt {a^2-b^2}-160\,a^2\,b^2\,{\mathrm {e}}^x+96\,a\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2-b^2}\right )\,\sqrt {a^2-b^2}}{a^2}-\frac {b\,\left (\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{2}{\relax (x )}}{a + b \cosh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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